Exceptional Sets of Slices for Functions From the Bergman Space in the Ball

http://dx.doi.org/10.4153/CMB-2001-019-7
Canad. Math. Bull. 44(2001), 150-159

  Published:2001-06-01
 Printed: Jun 2001

Let $B_N$ be the unit ball in $\mathbb{C}^N$ and let $f$ be a function holomorphic and $L^2$-integrable in $B_N$. Denote by $E(B_N,f)$ the set of all slices of the form $\Pi =L\cap B_N$, where $L$ is a complex one-dimensional subspace of $\mathbb{C}^N$, for which $f|_{\Pi}$ is not $L^2$-integrable (with respect to the Lebesgue measure on $L$). Call this set the exceptional set for $f$. We give a characterization of exceptional sets which are closed in the natural topology of slices.